Volume 88A, number2 PHYSICSLETTERS 22 February1982

EXPLODE-DECAY MODE LUMP SOLITONSOF A TWO-DIMENSIONAL NONLJNEAR SCHRODINGER EQUATION

Akira NAKAMURADepartmentofMathematicsand ComputerScience,ClarksonCollegeof Technology,Potsdam,NY136 76, USA

Received16 November1981

For a two-dimensionalnonlinearSchrodingerequation,a transformationhasbeenobtainedwhich transformusualpropagatingwavesolutionsto theexplode-decaymodesolutions.This is appliedto thelump solitonsandexplicit explode-decaymodelumpsolitonsolutionshavebeenobtained.

Thestudyof nonlinearevolutionequationshas satisfyeqs.(1), then U and JV satisfythe sameequa-beenattractinginèreasinginterestsrecently.In this tion with thesubscriptsx,y, t replacedby X, Y, T.paperwe considertwo-dimensional(2D) nonlinear This transformationhasbeenobtainedfrom the di-Schrodinger(NLS) equationwritten as [1—6] rectanalogyto thesimilar result [8] of the cubic2D

* NLS equationwritten as1U~+ (~

3)U~~+ yuyy + ~u uu 2wu = 0 , (la)+ + + 5U*UU = 0.

j3w +yw _13ö(u*u) 0. (lb)xx yy XX The presenttransformationgeneratesexplode-decay

Here andin thefollowing, the subscriptsx,y,t rep- modeone-solitonsfrom theordinaryone-solitons.Itsresentpartial derivativesandi3, y, ~ are arbitrarycon- relationis quiteparallelto the caseof the cubic 2Dstants.AnkerandFreeman[5] obtainedmultiple NLS equation[8] and is not replacedhere.If we startsoliton solutionsto this equationby using the inverse from the multiple soliton solutions,it generatesex-scatteringmethod(1ST). Multiple solitonsare the non- plode-decaymodemultiple solitonswhichwerepre-linear superpositionstateof iD-like solitonsin 2D ar- viously derivedby usingonly bilineartransformtech-bitrarypropagatingdirections.Recentlyby usingthe niques [7]. Sincethe transformationis independentbilinearmethod,wehaveshownthat thereexist sim- of the explicit form of thesolutions,it worksforple similarity-typeexplode-decaysolitons(ripplons) othermodesas well. Forexample,it generatesexplode-which are similar to the multiple solitonsmentioned decaymodemultiple periodicwavesfrom the corre-just abovebut withsolitonamplitudesgrowingand spondingordinarymultiple periodicwaves.It is alsodecayingwith time [7]. The purposeof this paperis known thateq. (1) admitsotherinterestingsolutionsto further studytheexplode-decay(ripplon) mode, suchas resonancestateof solitons[5] andlump soli-

For this purposeit is convenientto usethe follow- tons [6]. The presenttransformindicatesthat thereing transformation.We find bydirect calculationthat exist also explode-decayresonancesolitonsandex-if u andw,written as plode-decaylump solitons.

u(x,y, t) = t~ exp[ix2/4(—jl)t + iy2/4’ytl u(x,~ Herwe investigatethe latterin detail.Eq.(1) hasthelump one-solitonsolution

w(x,y,t) t2W(X, Y,fl,

X~x/t, Y~y/t, T~—1/t, (2)

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Volume88A, number2 PHYSICSLETTERS 22 lebruary 1982

u = g/f, w = (213 log~ , g = p~exp(O~) trast to theusual caseof aniplitude dependentvelocityof ordinary solitons.Time variationof thepeakampli-

f I + 8~’6p7p~[~x+x~+ 213k~t)2/13 tude ut (atx = 213k~-~ xj,v = 2yl~j’~t)is given byiti~ which grows(t <0), explodesat t 0, andthen

+ (y +y~ 2yl~t)2/y], decays(t> 0) andvanishesat t = o~•At any value oftime t, the profile of theabsoluteamplitude ui is of

i(k~x+l~y+ wet), yl~ (3) thehump with onepeakwhich decaysmonotonouslyto anydirection in space.

wherek~,~ (ps)arereal (complex)arbitrary For the multiple lump solitons, the sameargumentconstants.Thesolution (3) can be obtainedfor exam- goesin quite a parallelmanner.SatsurnaandAblowitz

pIe throughtheHirotabilinearmethodby settingu = [61consideredmultiple lump solitonsof thepresentg/f, w = (213 logJ)~~(f=f*) andrewriting (1) into bi- equation(with the nonvanishingboundaryconditions,linearform [6,7] u(x = ±oo)�:0). Thenapplicationof the transformation

~iD + ‘—~‘D2+ D2’ ~ 0 (2) to their ordinary multiple lump solitonsgenerates~‘i x ~‘ y1~“ — ‘ multiple explode-decaylump solitons.(i?D2 + D2~~•~‘ 6 * = 0 Finally, thepresenttransformationitself doesnot~ x ~‘ y~ ~ — g g (4) tell us anythingaboutthe superpositionof ordinary

andcheckingthat thesolution g,f givenby (3) actual- solutionsandthe explode-decaymodesolutions.How-ly satisfieseq.(4). HereoperatorsD~,D~... areHirota ever,knowing theexplicit form of explode-decaymodebilinear derivativeoperatorsrepresentingD~g.fEg~f solutions,it is not difficult to incorporatethesesolu-

— ~ — 2g~f~+gf~~andso on. For tions into, say, the 1ST formalismof eq. (1) calculatedbrevity hereafterwe assume13, y, 6 all positive.Eq. (3) by Anker andFreenian.Thenfrom thelinearity of theshowsthat iu i hastheprofile of ahump with its peak 1ST scheme,it canbe easilyshownthat thesuperposi-locatedat x = — 213ktt,y = —ye + 2-yl~twhich tion of solitons andripplons arepossible[91.monotonouslydecaysto any directionin spaceand

moveswith constantvelocity (—2i3k~,2yl~).By apply- Thepresentwork waspartiallysupportedby theing thepresenttransform(2) to lump one-soliton(3), AFOSRandONR.we havean explode-decaylump one-soliton(lump one-

ripplon) written as References

u =g/f, w = (2fllogf~~~ [11 D.J.BenneyandG.J. Roskes,Stud. AppI. Math.48

(1969)377.g = t1 exp[ix2/4(—13)t+ iy2/4yt] [2] A. DaveyandK. Stewartson,Proc. R. Soc. A338 (1974)

101.

X p,.exp[i(k~.x+ 1~y— w~)/t], [3] NC. FreemanandA. Davey,Proc.R. Soc. A344 (1975)427.

* 2 [41M.J. Ablowitz andR. Haberman,Phys.Rev. Lett. 35f 1 + 8~p~p~6[~x 213kg +x~t)/13t2 (1975) 1185.

[5] D. Ankerand N.C. Freeman,Proc. R. Soc.A360 (1978)+(y +2yl~+y

1t)2/yt2J. (5) 529.

[6] J. SatsumaandM.J. Ablowitz, 1. Math.Phys.20 (1979)Fromeq.(5), we seethecharacteristicsof theprofile 1496.of a lump one-ripplonasfollows. As t —~±°°, ui —~-0. [7] A. Nakamura,J. Math.Phys.23 (1982), to be published.

The humpmoveswith arbitrary constantvelocity (—xi, [8] A. Nakamura,J. Phys.Soc.Japan.50 (1981) 2469.- - . . . [9] A. Nakamura,preprmt.

~ This arbitrarinessof thepropagatingvelocity isthesameastheusualone-ripplon[7,8] andmakescon-

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